The Warm-up for today's lesson includes two problems. Problem (1) asks students to deal with a new notation, so I will need to explain what it means. Part of the big idea of the Common Core transition is to give students the opportunity to make sense of new ideas on their own, before getting input (MP1). Today's warmup targets this math practice. Even if I end up having to explain the notation to the whole class in the same way I used to, if students have the chance to think about it on their own first, they will understand it better when I explain it to them.

The idea of this problem is that students can use the data table to help understand the approach statement, as long as they set up the inputs correctly. I will ask students:

How can we choose inputs for the data table that will help us see what happens to the outputs?

The eventual goal is for students to interpret these two approach statements using the concept of division. I want them to fully understand statements like:

When you divide 1 by a really big number, that is like sharing 1 cookie with millions of people. So each person will only get crumbs of the cookie and the result is a really small amount.

When you divide 1 by a really small number, that is like asking how many times that really tiny number fits into 1 whole. Or, if you have 1 cookie and you want to give 0.00001 cookies to each person, how many people can you give this part of your cookie to? Many many people—because this tiny number fits many times into 1 whole.

Even if it takes a lot of time, I think it is essential to have these conversations over and over again, because these two ideas are basically the core idea behind the asymptotes of rational functions. If students don’t understand these two ideas, they will never really understand asymptotes. Eventually, we will connect these two ideas directly to the horizontal and vertical asymptotes of the function y=1/x.

Problem (2) is designed to give students another opportunity to understand the function that relates the speed at which a person travels to the time it takes that person to cover a fixed distance. The problem today involves a wrinkle in which the person remains at their destination for a certain amount of time. This fixed amount of time becomes the horizontal asymptote of the rational function—which makes sense based on the context. No matter how fast you travel, your total trip time can never be less than the amount of time you spend at your destination. Framing it this way helps students make sense of the horizontal asymptote. To help my students get there, I ask them questions like:

What is the least possible time the trip can take you? Why does this make sense?

How does this show up on the graph?

What is the greatest possible time the trip can take you? Why does this make sense?

It is not necessary for students to answer all questions. The key idea is to reflect on the meaning of the asymptotes:

The horizontal asymptote represents what happens when the car lasts a million years; or when a billion people go to Prom; or when you travel 1,000 mph.

The vertical asymptote represents what happens to the time your trip takes when you travel 0.001 mph, or when the car lasts you only one day, or when you are the only person at Prom.

I use these exaggerated inputs to emphasize the idea that the value the outputs “approach” is what matters. We can be a little informal when thinking about asymptote.

I may have have students choose one set of questions, or answer them all in a group. The important thing is the whole-class debrief. I want to model how to think about these problems, because students are often not used to thinking about functions and relationships without any numbers. This is a great chance to reason abstractly (MP2) and eventually students will make connections between this abstract reasoning and the behavior of the graph of these functions.

I model this abstract reasoning through a “think-aloud” where I basically say one or two of the answers below aloud before then having students discuss the questions with a partner. Eventually I will ask the students to write down one or two answers so I can use this as a formative assessment, but it is important to remember that there is a gap between their understanding and what they are able to write down.